Haar Condition Research Articles

ЧЕБЫШЕВСКИЕ ПРОЕКЦИИ НА ПОЛИЭДР

The problem of minimizing weighted Chebyshev norm on a convex polyhedron defined as the set of solutions to a system of linear inequalities may have a non-unique solution. Moreover, among the solutions to this problem, there may be clearly not suitable points of the polyhedron for the role of the closest points to the zero vector. It complicates, in particular, the Chebyshev approximation. In order to overcome the problems arising from this, the Haar condition is used, which means the requirement for the uniqueness of the solution of the indicated problem. This requirement is not always easy to verify and it is not clear what to do if it is not true. An algorithm is presented that always generates a unique solution to the indicated problem, based on the search with respect to interior points for optimal solutions of a finite sequence of linear programming problems. The solution developed is called the Chebyshev projection of the origin onto the polyhedron. It is proved that this solution is a vector of a polyhedron with Pareto-minimal absolute values of the components. It is proved that the sets of Chebyshev (according to the introduced algorithm) and Euclidean projections of the origin of coordinates onto the polyhedron, formed by varying the positive weight coefficients in the minimized Euclidean and Chebyshev norms, coincide.

Chebyshev approximations do not need the Haar condition

Chebyshev approximations do not need the Haar condition

Random Phaseless Sampling for Causal Signals in Shift-Invariant Spaces: A Zero Distribution Perspective

We proved that the phaseless sampling (PLS) in the linear-phase modulated shift-invariant space (SIS) $V(e^{\textbf{i}\alpha \cdot}\varphi), \alpha\neq0,$ is impossible even though the real-valued function $\varphi$ enjoys the full spark property (so does $e^{\textbf{i}\alpha \cdot}\varphi$). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $\phi$ such that the PLS of nonseparable causal (NC) signals in $V(\phi)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability $1$, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability $1$, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.

Interior Point Method: History and Prospects

Two mutually dual families of interior point algorithms are considered. The history of creating the algorithms, the main theoretical results on their justification, the experience of practical use, possible directions of development, and methods for counteracting calculation errors are presented. Subsets of algorithms with various special properties are distinguished, including those that necessarily lead to relatively interior points of optimal solutions. An algorithm for finding the Chebyshev projection onto a linear manifold is presented, in which the properties of relatively interior points of optimal solutions are efficiently employed. This algorithm always elaborates a unique projection and allows one to dispense with the hard-to-verify and sometimes violated Haar condition.

An Analog of the Haar Condition for Simple Partial Fractions

We prove that for a continuous real-valued function f on the segment [−1, 1] a real-valued simple partial fraction R n with n distinct poles outside the unit disk is a fraction of degree at most n of best approximation and is unique if and only if for the difference f − R n on [−1, 1] there exists a Chebyshev alternance of n + 1 points. The result is applied to the problem on approximation of real constants.

On lower semi-continuous metric projections onto finite dimensional subspaces of spaces of continuous functions

The context of the paper is: a locally compact Hausdorff space T; the space C0(T), equipped with the uniform norm, of real continuous functions on T which vanish at infinity; a linear subspace G, of finite dimension n, of the space C0(T); and the set-valued metric projection PG of C0(T) into the family of non-empty compact convex subsets of G, defined by PG(f)= the set of best uniform approximations to f from G. Those G which are Chebyshev (that is the metric projection is single point valued) were characterised by Haar (1918). Those G for which the metric projection PG is lower semi-continuous have been characterised by Wu Li (1989) and A.L. Brown (2005); the paper interprets and exploits the characterisation. A ‘Generalised Haar Condition’ which is necessary for PG to be lower semi-continuous is identified; in some circumstances it is also sufficient. The condition has a calculable determinantal form. The results of the paper include an essentially complete determination, in case T is a compact space in which no net of components is convergent to a single point set, of those subspaces G of C(T) for which PG is lower semi-continuous. Simple examples show that if the space T is compact and totally disconnected, but not finite, the situation is essentially different.

Uniform approximation with radical functions

In this work we give a family of radical functions, C, uniformly dense in C [a, b]. This family does not satisfy the usual theorems of Approximation Theory about C [a, b]. The mathematical expression of the sequence that converges uniformly to every ƒ ∈ C[a, b] is determined. We present algorithms to approximate ƒ with the uniform norm in different practical instances. A linear subspace of radical functions that verifies the Haar condition is constructed. We also study the degree of uniform approximation by radical functions and we show a comparative analysis with the approximation by arbitrary polynomials and Bernstein polynomials.

The Basic Theorem and its Consequences

Let T be a compact Hausdor topological space and let M denote an n-dimensional subspace of the space C(T ), the space of real-valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii (7) attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel'man, which uses Helly's Theorem, now the paper obtains a new proof of the Basic Theorem. The significance of the Basic Theorem for us is that it reduces the characterization of a best approximation to f 2 C(T ) from M to the case of finite T , that is to the case of approximation in l 1 (r). If one solves the problem for the finite case of T then one can deduce the solution to the general case. An immediate consequence of the Basic Theorem is that for a finite dimensional subspace M of C0(T ) there exists a separating measure for M and f 2 C0(T )\M, the cardinality of whose support is not greater than dimM+1. This result is a special case of a more general abstract result due to Singer (5). Then the Basic Theorem is used to obtain a general characterization theorem of a best approximation from M to f 2 C(T ). We also use the Basic Theorem to establish the suciency of Haar's condition for a subspace M of C(T ) to be Chebyshev.

Chebyshev Design of Linear-Phase FIR Filters With Linear Equality Constraints

By using basis transformation, the Chebyshev approximation of linear-phase finite-impulse response (FIR) filters with linear equality constraints can be converted into an unconstrained one defined on a new function space. However, since the Haar condition is not necessarily satisfied in the new function space, the alternating property does not hold for the solution to the resulted unconstrained Chebyshev approximation problem. A sufficient condition for the best approximation is obtained in this brief, and based on this condition, an efficient single exchange algorithm is derived for the Chebyshev design of linear-phase FIR filters with linear equality constraints. Simulations show that the proposed algorithm can converge to the optimal solution in most cases and to a near-optimal solution otherwise. Design examples are presented to illustrate the performance of the proposed algorithm.

Best Approximation by Free Knot Splines

We consider the problem of finding the best (uniform) approximation of a given continuous function by spline functions with free knots. Our approach can be sketched as follows: By using the Gaus transform with arbitrary positive real parameter t, we map the set of splines under consideration onto a function space, which is arbitrarily elose to the spline set, but satisfies the local Haar condition and also possesses other nice structural properties. This enables us to give necessary and sufficient conditions for best approximations (in terms of alternants) and, under some assumptions, even full characterizations and a uniqueness result. By letting t -> 0, we recover a best approximation in the original spline space. Our results are illustrated by some numerical examples, which show in particular the nice alternation behavior of the error function.

Minimum norm design of two-dimensional weighted Chebyshev FIR filters

The weighted Chebyshev design of two-dimensional FIR filters is in general not unique since the Haar condition is not generally satisfied. However, for a design on a discrete frequency domain, the Haar condition might be fulfilled. The question of uniqueness is, however, rather extensive to investigate. It is therefore desirable to define some simple additional constraints to the Chebyshev design in order to obtain a unique solution. The weighted Chebyshev solution of minimum Euclidean filter weight norm is always unique, and represents a sensible additional constraint since it implies minimum white noise amplification. This unique Chebyshev solution can always be obtained by using an efficient quadratic programming formulation with a strictly convex objective function and linear constraints. An example where a conventional Chebyshev solution is nonunique is discussed.

NONLINEAR APPROXIMATION OF A VARIATION OF L1 UNDER THE LOCAL HAAR CONDITION

In this paper, we discuss the nonlinear approximation of a variation of L1 under the local Haar condition. Theorems on characterizations (including alternations), unicity and strong unicity of the nonlinear approximation are obtained, the associated linear problem equivalent to the nonlinear approximation are given.

A Practical Geometrically Convergent Cutting Plane Algorithm

In this paper we consider a practical variation of the Cheney–Goldstein–Kelley cutting plane algorithm which solves linear relaxations using the lexicographic dual simplex method and allows for the deletion of inactive cuts. The algorithm presented can be shown to exhibit a geometric global rate of convergence asymptotically under mild assumptions. Other proofs of the rate of convergence for this type of algorithm require restrictive assumptions such as strict convexity of the objective function or a Haar condition. The approach taken here requires only that at each iteration of the algorithm a cut is produced for each convex constraint.

Complex Chebyshev approximation for FIR filter design

The alternation theorem is at the core of efficient real Chebyshev approximation algorithms. In this paper, the alternation theorem is extended from the real-only to the complex case. The complex FIR filter design problem is reformulated so that it clearly satisfies the Haar condition of Chebyshev approximation. An efficient exchange algorithm is derived for designing complex FIR filters in the Chebyshev sense. By transforming the complex error function, the Remez exchange algorithm can be used to compute the optimal complex Chebyshev approximation. The algorithm converges to the optimal solution whenever the complex Chebyshev error alternates; in all other cases, the algorithm converges to the optimal Chebyshev approximation over a subset of the desired bands. The new algorithm is a generalization of the Parks-McClellan algorithm, so that arbitrary magnitude and phase responses can be approximated. Both causal and noncausal filters with complex or real-valued impulse responses can be designed. Numerical examples are presented to illustrate the performance of the proposed algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Uniform local strong uniqueness on finite subsets

LetF be an approximating function with parameter setP, a subset ofn-space, on an intervalI satisfying the hypotheses of Meinardus and Schwedt (the local Haar condition and propertyZ) which result in an alternating theory. Consider uniform approximation to functionf onX, a finite subset ofI. A sufficient condition is given, involving best parameters being in a closed set, on which the degree isn that a family of functionsf have a uniform (parameterwise) local strong uniqueness constant > 0. The necessity of the condition in this and related problems, in particular ordinary rational approximation on an interval, is examined.

Nonlinear Chebyshev fitting from the solution of ordinary differential equations

The nonlinear Chebyshev approximation of real-valued data is considered where the approximating functions are generated from the solution of parameter dependent initial value problems in ordinary differential equations. A theory for this process applied to the approximation of continuous functions on a continuum is developed by the authors in [17]. This is briefly described and extended to approximation on a discrete set. A much simplified proof of the local Haar condition is given. Some algorithmic details are described along with numerical examples of best approximations computed by the Exchange algorithm and a Gauss-Newton type method.

Generalized simultaneous Chebyshev approximation

In this note, we develop, without assuming the Haar condition, a generalized simulaneous Chebyshev approximation theory which is similar to the classical Chebyshev theory and contains it as a special case. Our results also contain those in [1] and [3] as a special case, and the two conjectures proposed by C. B. Dunham in [2] are proved to be true in the case of simultaneous approximation.

Discrete one-sided nonlinear Lp approximation

Under the local Haar condition, an interpolatory theorem for discrete one-sided nonlinear Lp (p>1) approximation is obtained. Under the strong Young’s condition, a Polya-type theorem with discretization for one-sided nonlinear approximation and a limit (discretization) theorem for one-sided modified Lp (p≥1) approximation are given.

A generalized Chebyshev theory

In this paper we develop, without assuming the Haar condition, a generalized Chebyshev theory for Chebyshev approximation which is similar to the classical Chebyshev theory and contains it as a special case.

Haar's condition for systems of vector-valued functions

Haar's condition for systems of vector-valued functions